The sum of two numbers is $74$, and their difference is $42$. What are the two numbers?
Solution: Let $x$ be the first number, and let $y$ be the second number. The system of equations is: ${x+y = 74}$ ${x-y = 42}$ Solve for $x$ and $y$ using elimination. Add the top and bottom equations together. $ 2x = 116 $ $ x = \dfrac{116}{2} $ ${x = 58}$ Now that you know ${x = 58}$ , plug it back into $ {x+y = 74}$ to find $y$ ${(58)}{ + y = 74}$ ${y = 16}$ You can also plug ${x = 58}$ into $ {x-y = 42}$ and get the same answer for $y$ ${(58)}{ - y = 42}$ ${y = 16}$ Therefore, the larger number is $58$, and the smaller number is $16$.